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Sagot :
To find the inverse of the square root parent function [tex]\( F(x) = \sqrt{x} \)[/tex], we need to determine which function, when composed with the square root function, results in the identity function [tex]\( x \)[/tex].
The definition of an inverse function [tex]\( g(x) \)[/tex] of [tex]\( f(x) \)[/tex] is such that if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( g \)[/tex] and [tex]\( f \)[/tex] respectively, then [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Here, [tex]\( F(x) = \sqrt{x} \)[/tex]. Let's consider the given options to see which function is its inverse.
### Option A: [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
- First, let's verify if this function is the inverse:
1. Given [tex]\( y = \sqrt{x} \)[/tex], squaring both sides yields [tex]\( y^2 = x \)[/tex]. So, [tex]\( \sqrt{x} \)[/tex] undoes [tex]\( x^2 \)[/tex] when [tex]\( x \geq 0 \)[/tex].
2. Similarly, if [tex]\( F(x) = x^2 \)[/tex] where [tex]\( x \geq 0 \)[/tex], then [tex]\( F^{-1}(x) = \sqrt{x} \)[/tex].
Thus, for [tex]\( y = \sqrt{x} \)[/tex], if [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x^2 \geq 0 \)[/tex].
- [tex]\( F^{-1}(F(x)) = x \)[/tex] and [tex]\( F(F^{-1}(x)) = x \)[/tex].
So, [tex]\( F(x) = x^2 \)[/tex], [tex]\( x \geq 0 \)[/tex] correctly defines [tex]\( \sqrt{x} \)[/tex] as its inverse.
### Option B: [tex]\( F(x) = |x| \)[/tex]
- This function maps both positive and negative [tex]\( x \)[/tex] to their absolute values. However, the inverse of [tex]\( \sqrt{x} \)[/tex] should map back to a specific single value.
- The square root function is not the inverse of the absolute value function, because [tex]\( \sqrt{x} \)[/tex] does not undo [tex]\( |x| \)[/tex].
### Option C: [tex]\( F(x) = \frac{1}{\sqrt{x}} \)[/tex]
- This option represents a reciprocal transformation, which is not related to squaring or unsquaring a number.
- The square root function is not the inverse of the reciprocal of the square root.
### Option D: [tex]\( F(x) = x^2 \)[/tex]
- Without specifying the domain restriction [tex]\( x \geq 0 \)[/tex], the function [tex]\( F(x) = x^2 \)[/tex] includes negative values.
- For negative values, [tex]\( \sqrt{x} \)[/tex] is not defined, so [tex]\( F(x) = x^2 \)[/tex] without the domain restriction cannot be correct.
### Conclusion
Thus, the correct function that [tex]\( F(x) = \sqrt{x} \)[/tex] is the inverse of is:
A. [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
The definition of an inverse function [tex]\( g(x) \)[/tex] of [tex]\( f(x) \)[/tex] is such that if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( g \)[/tex] and [tex]\( f \)[/tex] respectively, then [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Here, [tex]\( F(x) = \sqrt{x} \)[/tex]. Let's consider the given options to see which function is its inverse.
### Option A: [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
- First, let's verify if this function is the inverse:
1. Given [tex]\( y = \sqrt{x} \)[/tex], squaring both sides yields [tex]\( y^2 = x \)[/tex]. So, [tex]\( \sqrt{x} \)[/tex] undoes [tex]\( x^2 \)[/tex] when [tex]\( x \geq 0 \)[/tex].
2. Similarly, if [tex]\( F(x) = x^2 \)[/tex] where [tex]\( x \geq 0 \)[/tex], then [tex]\( F^{-1}(x) = \sqrt{x} \)[/tex].
Thus, for [tex]\( y = \sqrt{x} \)[/tex], if [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x^2 \geq 0 \)[/tex].
- [tex]\( F^{-1}(F(x)) = x \)[/tex] and [tex]\( F(F^{-1}(x)) = x \)[/tex].
So, [tex]\( F(x) = x^2 \)[/tex], [tex]\( x \geq 0 \)[/tex] correctly defines [tex]\( \sqrt{x} \)[/tex] as its inverse.
### Option B: [tex]\( F(x) = |x| \)[/tex]
- This function maps both positive and negative [tex]\( x \)[/tex] to their absolute values. However, the inverse of [tex]\( \sqrt{x} \)[/tex] should map back to a specific single value.
- The square root function is not the inverse of the absolute value function, because [tex]\( \sqrt{x} \)[/tex] does not undo [tex]\( |x| \)[/tex].
### Option C: [tex]\( F(x) = \frac{1}{\sqrt{x}} \)[/tex]
- This option represents a reciprocal transformation, which is not related to squaring or unsquaring a number.
- The square root function is not the inverse of the reciprocal of the square root.
### Option D: [tex]\( F(x) = x^2 \)[/tex]
- Without specifying the domain restriction [tex]\( x \geq 0 \)[/tex], the function [tex]\( F(x) = x^2 \)[/tex] includes negative values.
- For negative values, [tex]\( \sqrt{x} \)[/tex] is not defined, so [tex]\( F(x) = x^2 \)[/tex] without the domain restriction cannot be correct.
### Conclusion
Thus, the correct function that [tex]\( F(x) = \sqrt{x} \)[/tex] is the inverse of is:
A. [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
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